I got into a discussion with my brother yesterday about the best feats for a planned archer alchemist. I had planned on going for Focused Shot and Deadly Aim, since I'd have about +5 from INT, to get big damage on a single arrow. He suggested dropping Deadly Aim in favor of Rapid Shot, so I could choose between a single big arrow (for keeping my move action, or readying to disrupt spellcasters), or a full-attack for slightly higher DPR. But I figure if I go the full-attack route, I should go all out. What do you think?

Focused Shot only:
Pros - Less feat-intensive, leaving space for Weapon Focus, or non-archery feats. Higher damage for disrupting spellcasters or bypassing DR.
Cons - Doesn't scale well with level. Lower DPR when I could do a full-attack.

Pros: Don't waste a feat on something sucky, you get consistently higher damage every round, Rapid Shot can (and should) be used with Fast Bombs, DR won't matter because every archer should get Clustered Shots anyway.

Cons: You do about 5-7 less damage once in a blue moon when you actually need your move action, which should be extremely rare as an archer--oh, yeah, except you don't actually because you can just throw a bomb or drink an elixir or something.

Yeah, Focused Shot is awful. Please don't take it. Every time someone takes Focused Shot, a baby cries for no reason.

Hammer the Gap (Combat)
You repeatedly strike the same location, causing increasing amounts of damage.

Prerequisite: Base attack bonus +6.

Benefit: When you take a full-attack action, each consecutive hit against the same opponent deals extra damage equal to the number of previous consecutive hits you have made against that opponent this turn. This damage is multiplied on a critical hit.

Vital Strike (Combat)
You make a single attack that deals significantly more damage than normal.

Prerequisites: Base attack bonus +6.

Benefit: When you use the attack action, you can make one attack at your highest base attack bonus that deals additional damage. Roll the weapon's damage dice for the attack twice and add the results together before adding bonuses from Strength, weapon abilities (such as flaming), precision based damage, and other damage bonuses. These extra weapon damage dice are not multiplied on a critical hit, but are added to the total.

These two feat do both jobs with no feat tax and work range and melee as well. So you can swith hit.
I would also add in
Arcane Strike (Combat)
You draw upon your arcane power to enhance your weapons with magical energy.

Prerequisite: Ability to cast arcane spells.

Benefit: As a swift action, you can imbue your weapons with a fraction of your power. For 1 round, your weapons deal +1 damage and are treated as magic for the purpose of overcoming damage reduction. For every five caster levels you possess, this bonus increases by +1, to a maximum of +5 at 20th level.

Therefore, Rapid Shot and Deadly Aim has the highest damage potential. However, considering that you're taking a -3 on your attack rolls, it may be in your best interest to combine Focused Shot and Deadly Aim.

I'm playing an archer bard at the moment and currently have Rapid Shot as one of my feats. We're level 7 at the moment so I don't have any iterative attacks.

Several times over the course of the past sessions I have used my full attack action to fire with rapid shot giving me 2 shots instead of 1. With alarming regularity I have rolled poorly on my first attack roll and missed my target. Without rapid shot my turn would largely be over. Thanks to rapid shot I get a second chance and usually hit.